Finite Group Databasebeta

← #13148887

Subgroups — #13148887

Order 1728.

772 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721743C211yesmaximal, characteristic, complemented
2C3×#63457C211yesmaximal, characteristic
3C9×#890C211yesmaximal, characteristic, complemented
4#11720998C212yesmaximal, complemented
5#11720998C211yesmaximal, characteristic, complemented
6#11720981C212yesmaximal, complemented
7#11720975C212yesmaximal, complemented
8C4×Dic3×D9C211yesmaximal, characteristic, complemented
9#11720948C211yesmaximal, characteristic, complemented
10#11720946C212yesmaximal, complemented
11#11720945C211yesmaximal, characteristic, complemented
12#10594427—31nomaximal
13#10589793—31nomaximal
14C2×#6226C2211yescharacteristic
15C2×C3×D36C2212yes—
16C2×C3×D36C2211yescharacteristic
17C2×C3×C4×D9C2211yescharacteristic
18C2×Dic3×D9C2211yescharacteristic
19C2×Dic3×D9C2211yescharacteristic
20C2×Dic3×D9C2212yes—
21C3×C9×C4⋊C4C2211yescharacteristic
22#88534C2212yes—
23C4×C3⋊Dic9C2211yescharacteristic
24C4×C3⋊Dic9C2212yes—
25C3×(C2×D9)⋊C4C2212yes—
26C9×C4⋊Dic3C2211yescharacteristic
27C9×C4⋊Dic3C2212yes—
28C9×Dic3⋊C4C2212yes—
29C4×C9×Dic3C2211yescharacteristic
30(C2×D9)⋊Dic3C2212yes—
31(C2×D9)⋊Dic3C2212yes—
32(C2×D9)⋊Dic3C2212yes—
33(C2×D9)⋊Dic3C2212yes—
34Dic3⋊Dic9C2212yes—
35Dic3×Dic9C2211yescharacteristic
36Dic3×Dic9C2211yescharacteristic
37#64096—31no—
38C3×#895—31no—
39C3×#890D311yescharacteristic
40#63898—32no—
41#63898—31no—
42#63884—32no—
43#63878—32no—
44C4×D3×Dic3—31no—
45#63860—31no—
46#63858—32no—
47#63857—31no—
48C9×#125—31no—
49#63460—32no—
50#63459—31no—
51#63457D311yescharacteristic
52#63456—32no—
53#63455—31no—
54#63455—32no—
55#63454—32no—
56#63449—31no—
57#63449—61no—
58#63437—31no—
59C42×D9—31no—
60C22×C3×D9C2312yes—
61C22×C3×D9C2311yescharacteristic
62C2×C3×C4×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C2×C3⋊Dic9C2311yescharacteristic
67#6226—22no—
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2312yes—
71C2×C3×Dic9C2311yescharacteristic
72C3×D36—24no—
73C3×D36—22no—
74C3×C4×D9—22no—
75Dic3×D9—22no—
76Dic3×D9—22no—
77Dic3×D9—24no—
78#5691—91no—
79#5600—91no—
80C2×#634—31no—
81C2×C3×D12—31no—
82C2×C3×D12—32no—
83C2×C3×C4×D3—31no—
84C2×D3×Dic3—31no—
85C2×D3×Dic3—31no—
86C2×D3×Dic3—32no—
87C32×C4⋊C4C2×D311yescharacteristic
88#3750—32no—
89C4×C3⋊Dic3—31no—
90C4×C3⋊Dic3—32no—
91C3×(C2×D3)⋊C4—32no—
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×C4⋊Dic3C2×D312yes—
94C3×Dic3⋊C4C2×D312yes—
95C3×C4×Dic3C2×D311yescharacteristic
96Dic3⋊Dic3—32no—
97(C2×D3)⋊Dic3—32no—
98(C2×D3)⋊Dic3—32no—
99(C2×D3)⋊Dic3—32no—
100(C2×D3)⋊Dic3—32no—
101Dic32—31no—
102Dic32—31no—
103C2×D36C2×D311yescharacteristic
104C2×D36C2×D312yes—
105C2×C4×D9—31no—
106C2×C4×D9—31no—
107C2×C4×D9—32no—
108C2×C4×D9C2×D311yescharacteristic
109C2×Dic18—31no—
110C2×Dic18—61no—
111C9×C4⋊C4—21no—
112C9×C4⋊C4—32no—
113C9×C4⋊C4—31no—
114C9×C4⋊C4—32no—
115C9×C4⋊C4C2×D311yescharacteristic
116C42×C9—31no—
117(C2×D9)⋊C4—32no—
118(C2×D9)⋊C4—32no—
119(C2×D9)⋊C4—32no—
120(C2×D9)⋊C4—32no—
121(C2×D9)⋊C4C2×D312yes—
122C4⋊Dic9—32no—
123Dic9⋊C4—32no—
124Dic9⋊C4—62no—
125C4×Dic9—31no—
126C4×Dic9—32no—
127C4×Dic9—31no—
128C4×Dic9—31no—
129C4×Dic9—62no—
130C4×Dic9—61no—
131C22×C3×C9C2411yescharacteristic
132C2×C3×D9—21no—
133C2×C3×D9—21no—
134C2×C3×D9—22no—
135C2×C3×D9—22no—
136C2×C3×D9—22no—
137C2×C3×D9#4112yes—
138C3×C4×C9—22no—
139C3×C4×C9#4112yes—
140C3⋊Dic9—22no—
141C3⋊Dic9—21no—
142C3⋊Dic9#4112yes—
143C9×Dic3—21no—
144C9×Dic3—21no—
145C9×Dic3—22no—
146C3×Dic9—21no—
147C3×#125—31no—
148C3×#121—91no—
149#936—91no—
150#935—92no—
151D4×Dic3—92no—
152D4×Dic3—91no—
153#924—91no—
154C2×C4×Dic3—91no—
155#898—92no—
156#897—91no—
157#895—31no—
158#894—92no—
159#893—91no—
160#893—92no—
161#892—92no—
162#890D911yescharacteristic, complemented
163#887—91no—
164#887—91no—
165#887—181no—
166#879—92no—
167#877—92no—
168#875—91no—
169C42×D3—91no—
170C22×C3×D3—31no—
171C22×C3×D3—32no—
172C2×C32×C4C22×D312yes—
173C2×C32×C4C22×D311yescharacteristic
174C2×C3⋊Dic3—31no—
175C2×C3⋊Dic3—31no—
176C2×C3⋊Dic3—32no—
177#634—62no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3C22×D311yescharacteristic
180C2×C3×Dic3C22×D311yescharacteristic
181C2×C3×Dic3C22×D312yes—
182C3×D12—64no—
183C3×D12—62no—
184C3×C4×D3—62no—
185D3×Dic3—62no—
186D3×Dic3—64no—
187D3×Dic3—62no—
188C22×D9C22×D312yes—
189C22×D9C22×D311yescharacteristic
190C2×C4×C9—21no—
191C2×C4×C9—22no—
192C2×C4×C9—31no—
193C2×C4×C9—31no—
194C2×C4×C9—32no—
195C2×C4×C9C22×D312yes—
196C2×C4×C9C22×D311yescharacteristic
197C2×Dic9—31no—
198C2×Dic9—31no—
199C2×Dic9—32no—
200C2×Dic9—61no—
201C2×Dic9—62no—
202C2×Dic9—61no—
203C2×Dic9C22×D311yescharacteristic
204D36—24no—
205D36—22no—
206C4×D9—22no—
207C4×D9—62no—
208C4×D9—64no—
209C4×D9—62no—
210Dic18—64no—
211Dic18—62no—
212#529—271noSylow
213C2×C3×C9C2×#4111yescharacteristic
214C2×C3×C9C2×#4111yescharacteristic
215C2×C3×C9C2×#4111yescharacteristic
216C3×D9—22no—
217C3×D9—42no—
218C2×C3×D4—92no—
219C2×C3×D4—91no—
220C22×C3×C4—91no—
221C22×Dic3—92no—
222C22×Dic3—91no—
223C22×Dic3—91no—
224C2×D12—31no—
225C2×D12—32no—
226C2×C4×D3—31no—
227C2×C4×D3—91no—
228C2×C4×D3—91no—
229C2×C4×D3—92no—
230C2×Dic6—91no—
231C2×Dic6—91no—
232C2×Dic6—181no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—32no—
235C3×C4⋊C4—31no—
236C3×C4⋊C4—32no—
237C3×C4⋊C4C2×D911yescharacteristic
238C3×C4⋊C4D3211yescharacteristic
239C3×C22⋊C4—92no—
240C3×C42—31no—
241#217—92no—
242#217—92no—
243#217—92no—
244#217—92no—
245(C2×D3)⋊C4—32no—
246(C2×D3)⋊C4—92no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—92no—
250C4⋊Dic3—92no—
251C4⋊Dic3C2×D912yes—
252C4⋊Dic3C2×D911yescharacteristic
253Dic3⋊C4—92no—
254Dic3⋊C4—92no—
255Dic3⋊C4—92no—
256Dic3⋊C4—182no—
257Dic3⋊C4C2×D912yes—
258C4×Dic3—91no—
259C4×Dic3—91no—
260C4×Dic3—92no—
261C4×Dic3—91no—
262C4×Dic3—91no—
263C4×Dic3—92no—
264C4×Dic3—91no—
265C4×Dic3—91no—
266C4×Dic3—181no—
267C4×Dic3—182no—
268C4×Dic3C2×D911yescharacteristic
269C22×C32C23×D311yescharacteristic
270C2×C3×D3—32no—
271C2×C3×D3—61no—
272C2×C3×D3—61no—
273C2×C3×D3—62no—
274C2×C3×D3—62no—
275C2×C3×D3—62no—
276C32×C4—22no—
277C32×C4#23912yes—
278C3⋊Dic3—32no—
279C3⋊Dic3—62no—
280C3⋊Dic3—61no—
281C3×Dic3—21no—
282C3×Dic3—21no—
283C3×Dic3—22no—
284C3×Dic3—61no—
285C22×C9—21no—
286C22×C9C23×D311yescharacteristic
287C2×D9—21no—
288C2×D9—21no—
289C2×D9—22no—
290C2×D9—22no—
291C2×D9—22no—
292C2×D9#23712yes—
293C4×C9—22no—
294C4×C9—22no—
295C4×C9—42no—
296C4×C9—61no—
297C4×C9—61no—
298C4×C9—62no—
299C4×C9#23712yes—
300Dic9—21no—
301Dic9—32no—
302Dic9—62no—
303Dic9—64no—
304Dic9—62no—
305Dic9—61no—
306Dic9—121no—
307#126—272no—
308#125—31no—
309#124—271no—
310#123—272no—
311C22⋊Q8—271no—
312#121—91no—
313C4×Q8—271no—
314C4×D4—271no—
315C4×D4—272no—
316#117—272no—
317C2×C42—271no—
318C3×C9#52911yescharacteristic, complemented, Sylow
319C23×C3—91no—
320C23×C3—92no—
321C22×D3—31no—
322C22×D3—32no—
323C3×D4—184no—
324C3×D4—182no—
325C2×C3×C4—21no—
326C2×C3×C4—22no—
327C2×C3×C4—31no—
328C2×C3×C4—31no—
329C2×C3×C4—32no—
330C2×C3×C4—91no—
331C2×C3×C4—182no—
332C2×C3×C4C22×D912yes—
333C2×C3×C4C22×D911yescharacteristic
334C2×C3×C4C2×D3212yes—
335C2×C3×C4C2×D3211yescharacteristic
336C2×Dic3—31no—
337C2×Dic3—91no—
338C2×Dic3—91no—
339C2×Dic3—91no—
340C2×Dic3—92no—
341C2×Dic3—91no—
342C2×Dic3—92no—
343C2×Dic3—181no—
344C2×Dic3—182no—
345C2×Dic3—181no—
346C2×Dic3—182no—
347C2×Dic3—184no—
348C2×Dic3—182no—
349C2×Dic3C22×D911yescharacteristic
350C2×Dic3C22×D911yescharacteristic
351C2×Dic3C22×D912yes—
352D12—64no—
353D12—62no—
354C4×D3—62no—
355C4×D3—182no—
356C4×D3—184no—
357C4×D3—182no—
358Dic6—184no—
359Dic6—182no—
360Dic6—182no—
361C2×C32C2×#23911yescharacteristic
362C2×C32D3×#4111yescharacteristic
363C2×C32D3×#4111yescharacteristic
364C3×D3—62no—
365C3×D3—122no—
366C2×C9—21no—
367C2×C9—21no—
368C2×C9—21no—
369C2×C9C2×#23711yescharacteristic
370C2×C9C2×#23711yescharacteristic
371C2×C9D3×#4111yescharacteristic
372D9—22no—
373D9—42no—
374C2×Q8—271no—
375C2×D4—92no—
376C2×D4—91no—
377C22×C4—91no—
378C22×C4—271no—
379C22×C4—271no—
380C22×C4—272no—
381C4⋊C4—32no—
382C4⋊C4—31no—
383C4⋊C4—32no—
384C4⋊C4—272no—
385C4⋊C4—272no—
386C4⋊C4D3×D911yescharacteristic
387C22⋊C4—92no—
388C22⋊C4—272no—
389C22⋊C4—272no—
390C22⋊C4—272no—
391C22⋊C4—272no—
392C42—31no—
393C42—271no—
394C42—271no—
395C42—271no—
396C42—272no—
397C22×C3—21no—
398C22×C3—92no—
399C22×C3—181no—
400C22×C3—181no—
401C22×C3—182no—
402C22×C3—182no—
403C22×C3—182no—
404C22×C3C23×D911yescharacteristic
405C22×C3C22×D3211yescharacteristic
406C2×D3—32no—
407C2×D3—61no—
408C2×D3—61no—
409C2×D3—62no—
410C2×D3—62no—
411C2×D3—62no—
412C3×C4—22no—
413C3×C4—22no—
414C3×C4—22no—
415C3×C4—42no—
416C3×C4—61no—
417C3×C4—61no—
418C3×C4—62no—
419C3×C4—181no—
420C3×C4#370112yes—
421C3×C4#379612yes—
422Dic3—21no—
423Dic3—21no—
424Dic3—22no—
425Dic3—61no—
426Dic3—92no—
427Dic3—92no—
428Dic3—182no—
429Dic3—184no—
430Dic3—182no—
431Dic3—181no—
432Dic3—182no—
433Dic3—181no—
434Dic3—361no—
435C32#569111yescharacteristic
436C9—21no—
437C9#560011yescharacteristic, complemented
438C23—91no—
439C23—92no—
440Q8—542no—
441D4—184no—
442D4—182no—
443C2×C4—31no—
444C2×C4—31no—
445C2×C4—32no—
446C2×C4—91no—
447C2×C4—182no—
448C2×C4—271no—
449C2×C4—271no—
450C2×C4—272no—
451C2×C4—542no—
452C2×C4—544no—
453C2×C4—542no—
454C2×C4C2×D3×D912yes—
455C2×C4C2×D3×D911yescharacteristic
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—182no—
460C2×C3—362no—
461C2×C3C2×#370111yescharacteristic
462C2×C3#41×D911yescharacteristic
463C2×C3#41×D911yescharacteristic
464C2×C3C2×#379611yescharacteristic
465C2×C3D3×#23711yescharacteristic
466C2×C3#6431911yescharacteristic
467D3—62no—
468D3—122no—
469C22—92no—
470C22—181no—
471C22—181no—
472C22—182no—
473C22—182no—
474C22—182no—
475C22C22×D3×D911yescharacteristic
476C4—22no—
477C4—61no—
478C4—61no—
479C4—62no—
480C4—181no—
481C4—272no—
482C4—541no—
483C4—542no—
484C4#8863412yes—
485C3—21no—
486C3#1058979311yescharacteristic, complemented
487C3#1059442711yescharacteristic
488C2—182no—
489C2—362no—
490C2C2×#8863411yescharacteristic
491C2D9×#23711yescharacteristic
492C2#1172251411yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC3×#63457C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#529
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #11721743C2semidirect
3. C9×#890C2semidirect
4. #11720998C2semidirect
5. #11720998C2semidirect
6. #11720981C2semidirect
7. #11720975C2semidirect
8. C4×Dic3×D9C2semidirect
9. #11720948C2semidirect
10. #11720946C2semidirect
11. #11720945C2semidirect
162. #890D9semidirect
318. C3×C9#529semidirect
437. C9#5600semidirect
486. C3#10589793semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#117217432
C3×#634572
C9×#8902
#117209982
#117209812
#117209752
C4×Dic3×D92
#117209482
#117209462
#117209452
#105944273
#105897933